Rules of Inference
Dr. Yasir Ali
Syllogism
An argument form consisting of two premises and a conclusion is
called a syllogism.
The first and second premises are called the major premise and
minor premise, respectively.
Modus Ponens
The most famous form of valid syllogism in logic is called modus
ponens. It has the following form:
Following argument form is valid.
If the sum of the digits of 371,487 is divisible by 3, then 371,487
is divisible by 3.
The sum of the digits of 371,487 is divisible by 3.
Therefore, 371,487 is divisible by 3. p
q
p →q p
q
T
T
T
T
T
T
F
F
T
F
F
T
T
F
T
F
F
T
F
F
Modus Tollens
Another valid argument form called modus tollens.
It has the following form:
If p then q.
￢q
∴￢p
p
q
p →q ￢q
￢p
T
T
T
F
F
T
F
F
T
F
F
T
T
F
T
F
F
T
T
T
Following argument form is valid.
If the watch-dog detects an intruder, the dog will bark. The dog
did not bark. Therefore, no intruder was detected by the watchdog.
Additional Valid Argument Forms:
Rules of Inference
Generalization:
The following argument forms are valid.
Following argument form is valid.
“It is below freezing now”, therefore, “It is below freezing now”
or “It is raining now.”
Or simply,
It is below freezing now, therefore, it is below freezing or raining
now.
Also,
It is raining now, therefore, it is below freezing or raining now.
Specialization:
The following argument forms are valid.
Following argument form is valid.
“It is below freezing now,” and “It is raining now”, therefore, it is
below freezing now.
Or simply,
It is below freezing and raining now, therefore it is below
freezing now.
Also,
It is below freezing and raining now, therefore it is raining now.
Elimination:
The following argument forms are valid.
Following argument form is valid.
It rains today or we will have a barbecue today. We won’t have
barbecue today, therefore, it is raining today.
Also,
It rains today or we will have a barbecue today. It is not raining
today, therefore, we won’t have barbecue today.
Transitivity:
The following argument forms are valid.
Following argument form is valid.
If it rains today, then we will not have a barbecue today. If we do
not have a barbecue today, then we will have a barbecue
tomorrow. Therefore, if it rains today, then we will have a
barbecue tomorrow.
Fallacies
A fallacy is an error in reasoning that results in an invalid
argument.
Consider the following argument form:
If Zeke is a cheater, then Zeke sits in back row. Zeke sits in the
back row, therefore Zeke is a cheater.
The above argument has the form:
Which is not a valid argument form, this error in reasoning is
known as converse error.
p
q
p →q q p
T
T
T
T
T
T
F
F
F
T
F
T
T
T
F
F
F
T
F
F
Consider the following argument:
If interest rates are going up, stock market prices will go down.
Interest rates are not going up. Therefore, stock market prices
will not go down.
This is an invalid argument form, and the error in reasoning is
known as inverse error.
p
q
p →q
￢p
￢q
T
T
T
F
F
T
F
F
F
T
F
T
T
T
F
F
F
T
T
T
Rules of Inference
Rule of Inference
Tautology
Name
p
p→q
(p ∧ (p → q)) → q
Modus ponens
(￢q ∧ (p → q))→￢p
Modus tollens
((p → q) ∧ (q → r)) → (p
→r
) Hypothetical syllogism
((p ∨ q)∧￢p) → q
Disjunctive syllogism
p
∴p∨q
p → (p ∨ q)
Addition
p∧q
∴p
(p ∧ q) → p
Simplification
p
q
((p) ∧ (q)) → (p ∧ q)
Conjunction
((p ∨ q) ∧ (￢p ∨ r)) → (q
∨ r)
Resolution
￢q
p→q
p→q
q→r
p∨q
￢p
p∨q
￢p ∨ r
∴q
∴ ￢p
∴p→r
∴q
∴p∧q
∴q∨r
Example:
Show that the premises “If you send me an e-mail message,
then I will finish writing the program,” “If you do not send me an
e-mail message, then I will go to sleep early,” and “If I go to
sleep early, then I will wake up feeling refreshed” lead to the
conclusion “If I do not finish writing the program, then I will
wake up feeling refreshed.”
Let p :“You send me an e-mail message,”
q :“I will finish writing the program,”
r :“I will go to sleep early,” and
s :“I will wake up feeling refreshed.”
Then the argument takes the form
p → q,
￢p → r,
r→s
Therefore ￢q → s.
A set of premises and a conclusion are given. Use the valid
argument forms (rules of inference) to deduce the conclusion
from the premises, giving a reason for each step. Assume all
variables are statement variables.
1. p →q
2. r ∨ s
3. ￢ s → ￢ t
4. ￢ q ∨ s
5. ￢ s
6. ￢ p ∧ r → u
7.
8.
w∨t
∴u∧w
Given the following information about a computer program,
find the mistake in the program.
1. There is an undeclared variable or there is a syntax error in
the first five lines.
2. If there is a syntax error in the first five lines, then there is a
missing semicolon or a variable name is misspelled.
3. There is not a missing semicolon.
4. There is not a misspelled variable name.